# Difference between revisions of "ApCoCoA-1:NCo.HF"

(New page: <command> <title>NCo.HF</title> <short_description> Compute the values of the Hilbert function of a finitely generated <tt>K</tt>-algebra. <par/> For every <tt>i</tt> in <tt>N</tt>, we let...) |
m (replaced <quotes> tag by real quotes) |
||

(7 intermediate revisions by 3 users not shown) | |||

Line 1: | Line 1: | ||

+ | {{Version|1}} | ||

<command> | <command> | ||

<title>NCo.HF</title> | <title>NCo.HF</title> | ||

<short_description> | <short_description> | ||

− | + | Enumerate values of the Hilbert-Dehn function of a finitely generated <tt>K</tt>-algebra. | |

− | |||

− | |||

</short_description> | </short_description> | ||

<syntax> | <syntax> | ||

− | NCo.HF(Gb:LIST | + | NCo.HF(Gb:LIST[, DB:INT]):LIST |

− | |||

</syntax> | </syntax> | ||

<description> | <description> | ||

+ | Let <tt>K<X></tt> be a finitely generated non-commutative polynomial ring, and let <tt>I</tt> be a finitely generated two-sided ideal in <tt>K<X></tt>. Then <tt>K<X>/I</tt> is a finitely generated <tt>I</tt>-algebra. For every integer <tt>i</tt> in <tt>N</tt>, we let <tt>F_{i}</tt> be the <tt>K</tt>-vector subspace generated by the words of length less than or equal to <tt>i</tt>. Then <tt>{F_{i}}</tt> is a filtration of <tt>K<X></tt>. Further, the filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersects I)}</tt> of <tt>K<X>/I</tt>. The <em>Hilbert function</em> of <tt>K<X>/I</tt> is a map <tt>HF: N --></tt> N defined by <tt>HF(i)=dim(F_{i}/(F_{i} intersects I))-dim(F_{i-1}/(F_{i-1} intersects I))</tt>, i.e. <tt>HF(i)</tt> is equal to the number of words of length <tt>i</tt> in a Macaulay's basis (see <ref>ApCoCoA-1:NCo.MB|NCo.MB</ref>) of <tt>K<X>/I</tt>. | ||

+ | <par/> | ||

<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | <em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | ||

<par/> | <par/> | ||

− | Please set ring environment <em>coefficient field</em> <tt>K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before calling | + | Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before calling this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions. |

+ | <itemize> | ||

+ | <item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K<X></tt> forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt><X></tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. <em>Warning:</em> users should take responsibility to make sure that Gb is indeed a Groebner basis with respect to a length compatible word ordering! In the case that Gb is a partical Groebner basis, the function enumerates the values of a pseudo Hilbert function.</item> | ||

+ | |||

+ | <item>@return: a LIST of non-negative integers, which are values of the Hilbert function of the <tt>K</tt>-algebra <tt>K<X>/<Gb></tt>.</item> | ||

+ | </itemize> | ||

+ | Optional parameter: | ||

<itemize> | <itemize> | ||

− | <item>@param <em> | + | <item>@param <em>DB:</em> a positive INT, which is a degree bound of the Hilbert function. <em>Note that</em> we set DB=32 by default. Thus, in the case that the <tt>K</tt>-dimension of <tt>K<X>/<Gb></tt> is finite, it is necessary to set DB to a large enough INT in order to compute all the values of the Hilbert function.</item> |

− | |||

− | |||

</itemize> | </itemize> | ||

<example> | <example> | ||

− | NCo.SetX( | + | NCo.SetX("xyzt"); |

− | NCo.SetOrdering( | + | NCo.SetOrdering("LLEX"); |

− | Gb:= [[[1, | + | Gb:= [[[1, "yt"], [-1, "ty"]], [[1, "xt"], [-1, "tx"]], [[1, "xy"], [-1, "ty"]], [[1, "xx"], [-1, "yx"]], |

− | [[1, | + | [[1, "tyy"], [-1, "tty"]], [[1, "yyx"], [-1, "tyx"]]]; |

NCo.HF(Gb, 5); | NCo.HF(Gb, 5); | ||

[1, 4, 12, 34, 100, 292] | [1, 4, 12, 34, 100, 292] | ||

Line 30: | Line 34: | ||

</description> | </description> | ||

<seealso> | <seealso> | ||

− | <see>NCo.SetFp</see> | + | <see>ApCoCoA-1:NCo.IsGB|NCo.IsGB</see> |

− | <see>NCo.SetOrdering</see> | + | <see>ApCoCoA-1:NCo.MB|NCo.MB</see> |

− | <see>NCo.SetX</see> | + | <see>ApCoCoA-1:NCo.SetFp|NCo.SetFp</see> |

− | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see> |

+ | <see>ApCoCoA-1:NCo.SetX|NCo.SetX</see> | ||

+ | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> | ||

</seealso> | </seealso> | ||

<types> | <types> | ||

<type>apcocoaserver</type> | <type>apcocoaserver</type> | ||

+ | <type>ideal</type> | ||

<type>groebner</type> | <type>groebner</type> | ||

− | |||

<type>non_commutative</type> | <type>non_commutative</type> | ||

</types> | </types> | ||

Line 44: | Line 50: | ||

<key>NCo.HF</key> | <key>NCo.HF</key> | ||

<key>HF</key> | <key>HF</key> | ||

− | <wiki-category>Package_gbmr</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_gbmr</wiki-category> |

</command> | </command> |

## Latest revision as of 13:39, 29 October 2020

This article is about a function from ApCoCoA-1. |

## NCo.HF

Enumerate values of the Hilbert-Dehn function of a finitely generated `K`-algebra.

### Syntax

NCo.HF(Gb:LIST[, DB:INT]):LIST

### Description

Let `K<X>` be a finitely generated non-commutative polynomial ring, and let `I` be a finitely generated two-sided ideal in `K<X>`. Then `K<X>/I` is a finitely generated `I`-algebra. For every integer `i` in `N`, we let `F_{i}` be the `K`-vector subspace generated by the words of length less than or equal to `i`. Then `{F_{i}}` is a filtration of `K<X>`. Further, the filtration `{F_{i}}` induces a filtration `{F_{i}/(F_{i} intersects I)}` of `K<X>/I`. The *Hilbert function* of `K<X>/I` is a map `HF: N -->` N defined by `HF(i)=dim(F_{i}/(F_{i} intersects I))-dim(F_{i-1}/(F_{i-1} intersects I))`, i.e. `HF(i)` is equal to the number of words of length `i` in a Macaulay's basis (see NCo.MB) of `K<X>/I`.

*Please note:* The function(s) explained on this page is/are using the *ApCoCoAServer*. You will have to start the ApCoCoAServer in order to use it/them.

Please set ring environment *coefficient field* ` K`, *alphabet* (or set of indeterminates) `X` and *ordering* via the functions NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before calling this function. The default coefficient field is `Q`, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

@param

*Gb:*a LIST of non-zero polynomials in`K<X>`forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in`<X>`and C is the coefficient of W. For example, the polynomial`f=xy-y+1`is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].*Warning:*users should take responsibility to make sure that Gb is indeed a Groebner basis with respect to a length compatible word ordering! In the case that Gb is a partical Groebner basis, the function enumerates the values of a pseudo Hilbert function.@return: a LIST of non-negative integers, which are values of the Hilbert function of the

`K`-algebra`K<X>/<Gb>`.

Optional parameter:

@param

*DB:*a positive INT, which is a degree bound of the Hilbert function.*Note that*we set DB=32 by default. Thus, in the case that the`K`-dimension of`K<X>/<Gb>`is finite, it is necessary to set DB to a large enough INT in order to compute all the values of the Hilbert function.

#### Example

NCo.SetX("xyzt"); NCo.SetOrdering("LLEX"); Gb:= [[[1, "yt"], [-1, "ty"]], [[1, "xt"], [-1, "tx"]], [[1, "xy"], [-1, "ty"]], [[1, "xx"], [-1, "yx"]], [[1, "tyy"], [-1, "tty"]], [[1, "yyx"], [-1, "tyx"]]]; NCo.HF(Gb, 5); [1, 4, 12, 34, 100, 292] -------------------------------

### See also